Gaussian Elimination — Row-Reduce, Then Back-Substitute

Watch forward elimination pick a pivot, eliminate a column, advance, then back-substitute to read off the solution of A·x = b. Edit the matrix and try your own system.

forward step 0

Augmented matrix [ A | b ]

pivot pivot row pivot column row being eliminated

Progress

Press Step to begin forward elimination.
We solve A·x = b by reducing [A|b] to row-echelon form (forward elimination with partial pivoting), then climbing back up (back substitution).
What you are watching, one step at a time:
  1. Pick pivot — in the current column, highlight the largest-magnitude entry (partial pivoting).
  2. Swap — bring that pivot row into position (this flips the determinant's sign).
  3. Eliminate — subtract a multiple of the pivot row from each row below so the column entry becomes 0.
  4. Advance — move to the next column and repeat until the matrix is upper-triangular (row-echelon form).
  5. Back-substitute — solve the last row for one unknown, then climb upward filling in the rest.
Use Edit matrix to enter your own A and b (last column is b), then Apply edits to re-solve.
Self-contained visualization over the reals with partial pivoting. The pivot is the largest-magnitude entry in the current column. See junior.md for the procedure and professional.md for the proof that row operations preserve the solution set.